Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Generalized Born-Jordan Distributions and Applications to the Reduction of Interferences

Prof. Dr. Elena Cordero talk

**Date:** 05.12.18 **Time:** 16.15 - 17.45 **Room:** ETH HG E 1.2

One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature, however, causes the appearance of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them are members of the Cohen class. Among them, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called "ghost frequencies" are damped quite well, and the noise is, in general, reduced. The very insight relies on the kernel of such a distribution, which contains the \emph{sinus cardinalis} $\mathrm{sinc}$, the Fourier transform\, of the first B-Spline $B_{1}$. Replacing the function $B_{1}$ with the spline or order $n$, denoted by $B_{n}$, yields the function $(\mathrm{sinc})^{n}$ on the Fourier side, whose decay at infinity increases with $n$. The related Cohen's Kernel is given by $\Theta^{n}(z_{1},z_{2})=\mathrm{sinc}^{n}(z_{1}\cdot _{2})$, $n\in\bN$. We study properties of the time-frequency distribution, called \emph{generalized Born-Jordan distribution of order $n$}, arising from these new kernels. Such representations display a great capacity of damping interferences and the reduction increases with $n$. This talk will show the different facets of this phenomenon, from visual comparisons to rigorous mathematical explanations. This is a joint work with Monika D\"{o}rfler, Maurice de Gosson (University of Vienna) and Fabio Nicola (Politecnico di Torino).